MECH 322 Instrumentation

Lab 10 Damped Vibration of a Weighted Cantilever Beam

Performed: 4/1/15

Group 0Miles Greiner

Lab Instructors:Marissa Tsugawa

ABSTRACT• The goal of this experiment was to measure the

dependence of oscillatory frequency of a weighted cantilever beam on the beam length, and compare it with predictions based on measured beam dimensions, elastic modulus and masses.

• The oscillations were measured using an accelerometer and computer data acquisition system.

• The frequency decreases as the beam length increased. The predicted frequencies were roughly 15% larger than the measured values, and their confidence intervals did not overlap.

• The damping rate was not constant during the experiment, and had little effect on the beam frequency.

Figure 1 VI Front Panel

Figure 2 VI Block Diagram

Formula Formula: v*1000/c

Statistics Statistics This Express VI produces the following measurements: Time of Maximum

Spectral Measurements Selected Measurements: Magnitude (Peak) View Phase: Wrapped and in Radians Windowing: Hanning Averaging: None

Table 1 Measured Steel Beam Properties

• The value and uncertainty in E were determined in Lab 5• W and T were measured using micrometers whose

uncertainty were determined in Lab 5.• LT, and LE were measured using a tape measure

– readability = 1/16 in• MT and MW were measured using a analytical scale

– readability = 0.1 g

Units Value3 s

UncertaintyElastic Modulus, E GPa 206.3 7.5

Beam Width, W inch 0.9968 0.0015Beam Thickness, T inch 0.1220 0.0015

Beam Total Length, LT inch 23.813 0.063End Length, LE inch 0.469 0.063

Beam Mass, MT g 377.2 0.1Combined Mass, MW g 333.9 0.1

• Acceleration versus time measurements were acquired at a sampling rate of 600 Hz for two beam lengths, LB = 7 and 13 inches.

• The sampling rate was sufficiently high so the peaks decreased monotonically

• The longer beam clearly had fewer peaks and a lower oscillatory frequency.

Figure 3 Acceleration versus Time

Figure 4 Oscillatory Amplitude Versus Frequency

• The sampling time and frequency were T1 = 10 sec and fS = 600 Hz, so the system is capable of detecting frequencies between 0.1 and 300 Hz, with a resolution of 0.1 Hz.

• For beam lengths of LB = 13 and 7 inches, the peak frequencies are, respectively, fM = 7.50 ± 0.1 and 18.60 ± 0.1 Hz. – These frequencies are easily detected from this plot.

Fig. 5 Peak Acceleration versus Time

• The average exponential decay constants for the beam lengths of LB = 13 and 7 inches, are b = -0.176 and -0.192, respectively

• The magnitude of these constants (slope of the curves) decreased slightly with time.

Table 2 Calculated Values and Uncertainties

• The intermediate mass is small compared to the equivalent mass. • For both beam lengths, the damping is sufficiently low so that the predicted

undamped and damped frequencies, fOP and fP, are nearly the same

• The predicted damped frequencies are roughly 15% higher than the measured values, and their confidence intervals do not overlap.

Value 3s Uncertainty Value 3s

Uncertainty

Beam Length, LB [m] 0.1778 0.0016 0.3302 0.0016Intermedate Mass, MI [kg] 0.033 0.001 0.055 0.001

Equivalent Mass, MEQ [kg] 0.367 0.001 0.389 0.001Equivalent Beam Spring

Constant, kEQ[N/m] 6912 402 1079 58

Predicted Undamped Frequency, foP

[Hz] 21.8 0.6 8.4 0.2Measured Damped

Frequency, fM [Hz] 18.60 0.05 7.50 0.05

Decay Constant, b [1/sec] -0.192 - -0.176 -Damping Coefficient, lM [Ns/m] 0.1409 - 0.1368 -

Damped Frequency, fp [Hz] 21.8 0.6 8.4 0.2Percent Difference

(fP/fM-1)*100% 17% - 12% -

LB = 7 inch LB = 13 inch

Units

Extra (not part of report)

Warning: Be careful to check your data before processing

• For example, see oscillations between 2 and 4 seconds

Very repeatable frequency for same beam length (7 and 15 inch)

• 7 inch

Class Summary 2006Measure versus Predicted Frequency

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25 30 35 40

fM [Hz]

f P [H

z]

Damping Coefficient

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50

fP [Hz]

CM

[Ns/

m]